That's a good idea if you have any half duplex stuff, as the Ethernet standards committee removed the requirement to support half duplex links several years ago, and vendors have been shipping switches that don't support it.
An Extron RGB-HDMI 300(A) is usable in a pinch for IIgs video, which is at the extreme low end of the range of frequencies the RGB-HDMI 300 will handle. The best I was able to do was to display the IIgs video with some artifacts at the edge of the display, but it could be used to verify the IIgs...
Yes, that appears suitable. Here is a link to a good description of what the X and Y capacitor classes mean. https://www.allaboutcircuits.com/technical-articles/safety-capacitor-class-x-and-class-y-capacitors/
Yes it is okay to use a higher voltage rating replacement.
If the replacement for the Y type caps are also Y type caps then you can use a different type. If the ceramic capacitor is not Y rated then you can not use it as a replacement. Since you didn't specify the manufacturer's capacitor...
The supply voltages look to be within normal tolerances, note that until there was some load the secondary supply voltages (i.e. everything that isn't +5 V) weren't quite as close, but under load they end up being more close to the targets.
Your problem lies elsewhere, possibly with some bad...
The first way is to disconnect the power supply and use a meter to check the unloaded voltages by probing the pins of the connector from the side that connects to the pins on the motherboard. The second is to probe the rear of the connector while it is under load while attached to the...
Also consider reseating the chips that are in DIP sockets. You might also press on the VSC leadless chip at UH2 to make sure it is seated (that style chip really needs the purpose built chip extractor to safely remove it). Also if you haven't already done so, reseat any interface cards and...
Verify that the power supply voltages are within tolerance. Also look if the clock battery that has leaked over the PCB. Which revision IIgs do you have?
Your tantalum capacitor likely just failed itself, it's a known issue. The bugs had nothing to do with it unless they left corrosive fluid on the circuit board that contributed to the failure condition.
Is it 50/60 Hz hum? Maybe you have other bad solder connections, possibly on the ground plane or a marginal filter capacitor on the primary side of the switching power supply.